For a lot of guys (myself included), a big fear in approaching is freezing up and not knowing what to say. So I'm gonna leave an example approach that I fall back on. Let me know what you do, and how I can improve mine.

Scene: dining hall, coffee place, where ever.

After spotting a HB alone, slowly walk toward her.

Me: "Hi, sorry to bother you, but you're really cute and I'd be kicking myself all day if I didn't come say hi to you"

If she responds with something negative or insulting, say "Well you were cute until you opened your mouth. Have a good day!"

If she responds positively or neutrally, I continue:

Me: "I'm Doriangreat by the way. What's your name?" Put hand out to shake

When she tells me her name without giving signals that she's uncomfortable, I move on to say:

Me: "I'm not a huge fan of eating/drinking alone. Do you mind if I sit with you?"

If she agrees, here are some questions I use to open the conversation:

-What's your story?

-I'm not usually this outgoing. How about you?

-Any hobbies?

-What's your major/job?

-Any family?

A lot of room for improvement. Any suggestions?

Edit: Got rid of the apology. Really looking for stuff to add to the approach, not just subtract from it. It usually goes over really well until I get to the conversation, so a transition to opening a good conversation thread would be nice.